Spin-affected reflexive and stretching separation of off-center droplet collision

Recent studies have demonstrated the significant roles of droplet self-spin motion in affecting the head-on collision of binary droplets. In this paper, we present a computational study by using the volume-of-fluid method to investigate the spin-affected droplet separation of off-center collisions, which are more probable in reality and phenomenologically richer than head-on collisions. Different separation modes are identified through a parametric study with varying spinning speed and impact parameter. A prominent finding is that increasing the droplet spinning speed tends to suppress the reflexive separation and to promote the stretching separation. Physically, the reflexive separation is suppressed because the increased rotational energy reduces the excessive reflexive kinetic energy within the droplet, which is the cause for the droplet reflexive separation. The stretching separation is promoted because the increased droplet angular momentum enhances the local stretching flow within the droplet, which tends to separate the droplet. The roles of orbital angular momentum and spin angular momentum in affecting the droplet separation are further substantiated by studying the collision between two spinning droplets with either the same or opposite chirality. In addition, a theoretical model based on conservation laws is proposed to qualitatively describe the boundaries of coalescence-separation transition influenced by droplet self-spin motion.

Figure 1

Schematic of droplet collision regimes in a We-$B$ parameter space and experimental images for reflexive and stretching separation [4] for collision of identical droplets in atmosphere.

Figure 2

Comparison of (a) experimental images from Qian and Law [5], (b) simulation results from Chen et al. [42], and (c) the present results, for the nearly head-on separation of identical droplets at $\mathrm{We}=61.4$, $\mathrm{Oh}=0.028$, and $B=0.06$. The dimensionless time $T=t/t_{\mathrm{osc}}$ and $t_{\mathrm{osc}}={\left({\rho }_l{R_l}^3/\sigma \right)}^{1/2}=1.06\mathrm{ms}$.

Figure 3

Three-dimensional computational setup for an off-center collision between a spinning droplet $O_1$ and a nonspinning droplet $O_2$.

Figure 4

Evolution of droplet deformation upon a head-on collision of (a) two nonspinning droplets and of a spinning droplet $O_1$ and a nonspinning droplet $O_2$ with varying azimuthal angle of droplet $O_1$ (b) $\phi =0$, (c) $\phi =\pi /4$, and (d) $\phi =\pi /2$, at $\mathrm{We}=61.4$, $\mathrm{Oh}=0.028$, and ${\omega }_0=3$.

Figure 5

Collision between a spinning droplet and a nonspinning droplet with varying spinning speed ${\omega }_0$ and impact parameter $B$ at $\mathrm{We}=61.4$ and $\mathrm{Oh}=0.028$.

Figure 6

Contour of the viscous dissipation rate (VDR) and streamlines for the three cases of reflexive separation at $B=0.0$ that are shown in Figs. 5 with (a) ${\omega }_0=0$, (b) ${\omega }_0=3$, and (c) ${\omega }_0=6$.

Figure 7

Contour of the vorticity and kinetic energy (KE) fields and streamlines for the three cases of stretching separation at $B=0.4$ that are shown in Figs. 5 with (a) ${\omega }_0=0$, (b) ${\omega }_0=3$, and (c) ${\omega }_0=6$.

Figure 8

Contour of the vorticity field and satellite droplet formation upon the three cases of nearly grazing collision at $B=0.9$ that are shown in Figs. 5 with (a) ${\omega }_0=0$, (b) ${\omega }_0=3$, and (c) ${\omega }_0=6$.

Figure 9

Influence of chirality of droplet spin on the transition of coalescence-separation boundary at $\mathrm{Oh}=0.028$ for the case of (a) two nonspinning droplets and the cases of one spinning droplet with (b) ${\mathit{\omega }}_0\widehat{y}=-3$ and (c) ${\mathit{\omega }}_0\widehat{y}=3$.

Figure 10

Energy budget of the cases for the nearly head-on collision at $\mathrm{We}=61.4$ and $B=0.15$ that are extracted from Fig. 9, in which the total energy (TE), the surface energy (SE), the kinetic energy (KE), the total viscous dissipation energy (TVDE), and the total viscous dissipation rate (TVDR) are all nondimensional.

Figure 11

Contour of the vorticity field and streamlines for droplet reflexive separation that are shown in Fig. 10 for the case of (a) two nonspinning droplets and the cases of one spinning droplet with (b) ${\mathit{\omega }}_0\widehat{y}=-3$ and (c) ${\mathit{\omega }}_0\widehat{y}=3$.

Figure 12

Droplet deformation and vorticity contour for a head-on collision between two spinning droplets at $\mathrm{We}=61.4$, $\mathrm{Oh}=0.028$, and ${\omega }_0=3$. The spin motions have the (a) opposite direction and (b) the same direction.

Figure 13

Droplet deformation and vorticity contour for an off-center collision between two spinning droplets at $\mathrm{We}=61.4$, $\mathrm{Oh}=2.8\times {10}^{-2}$, $B=0.4$, and ${\omega }_0=3$. The spin motions have (a) opposite direction (droplet spin of $O_1$ is clockwise), (b) the same direction but droplet spin of $O_1$ is clockwise, and (c) the same direction but droplet spin of $O_1$ is anticlockwise.

Figure 14

Comparison of the predicted boundary for separations upon the collision between a spinning and a nonspinning droplet in Fig. 9. For sensitivity analysis, different values of ${\beta }_5$ and ${\beta }_6$ in large ranges are shown as (a) ${\beta }_5$ and (b) ${\beta }_6$ for $L_{\mathrm{t}}=L_{\mathrm{o}}+L_{\mathrm{s}1}$; (c) ${\beta }_5$ and (d) ${\beta }_6$ for $L_{\mathrm{t}}=L_{\mathrm{o}}-L_{\mathrm{s}1}$.

Figure 15

Comparison of the predicted boundaries of three representative cases in Fig. 9, (a) two nonspinning droplets, (b) ${\mathit{\omega }}_0\widehat{y}=-3$, and (c) ${\mathit{\omega }}_0\widehat{y}=3$.

Figure 16

Schematic of the model adopted in analyzing droplet reflexive and stretching separation for the collision between two nonspinning droplets.

Figure 17

Predicted boundary for the reflexive and stretching separation of two nonspinning droplets in the parameter space $\mathrm{We}\left(1-B^2\right)\sim \mathrm{We}{B}^2$. The dashed curve indicates the occurrence of droplet coalescence, which cannot be predicted by the present model. The criteria of Brazier-Smith et al. [1] and Al-Dirawi et al. [48] are also plotted for comparison.

Figure 18

Comparison of the predicted boundaries with different proportionality factors: (a) ${\beta }_2$, (b) ${\beta }_3$, and (c) ${\beta }_4$.